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On a problem of Chowla and some related problems. (English) Zbl 0015.24603

The problem in question is whether the integers \(n\) for which \(d(n+1)>d(n)\) have density \({1 \over 2}\), where \(d(n)\) denotes the number of divisors of \(n\). The author proves that this is the case. He first proves a general theorem to the effect that if \(f(n)\) satisfies (1)\( f(n) \geq 0\), (2) \(f(mn) = f(m)+f(n)\) provided that \((m,n) = 1\), (3) \(\sum f(p)/p\) (summed over all primes \(p\)) converges, then the density of the integers \(n\) for which \(f(n+1)<f(n)\) is \({1\over 2}\). The method of proof is based on that used by the author in a previous paper (see Zbl 0012.01004). The fundamental idea is that of approximating to \(f(n)\) by \[ f_k(n) = \sum_{p<p_k} f(p^\alpha), \text{ where } p^\alpha \mid m, p^{\alpha +1} \nmid m. \]
The author then establishes that the result of the theorem holds also for \(V(n)\), the number of different prime factors of \(n\), which satisfies (1) and (2) but not (3). The proof of this is on the same lines but much more complicated, as \(k\) is taken to be a function of \(n\) of the order of magnitude \(n^{(\log\log n)^{-3}}\). Finally the result for \(d(n)\) is obtained from that for \(V(n)\) by proving that for almost all \(n\), \[ (d(n+1)-d(n)) (V(n+1)-V(n)) > 0. \]
Reviewer: H. Davenport

MSC:

11N25 Distribution of integers with specified multiplicative constraints

Citations:

Zbl 0012.01004
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